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Properties

Label 2-2240-140.139-c1-0-66
Degree $2$
Conductor $2240$
Sign $0.314 + 0.949i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.86i·3-s + (2.07 − 0.836i)5-s + (2.64 + 0.168i)7-s − 0.486·9-s + 4.26i·11-s − 5.57·13-s + (−1.56 − 3.87i)15-s + 3.01·17-s − 0.0464·19-s + (0.314 − 4.93i)21-s + 5.58·23-s + (3.60 − 3.46i)25-s − 4.69i·27-s − 1.21·29-s − 4.62·31-s + ⋯
L(s)  = 1  − 1.07i·3-s + (0.927 − 0.374i)5-s + (0.997 + 0.0635i)7-s − 0.162·9-s + 1.28i·11-s − 1.54·13-s + (−0.403 − 0.999i)15-s + 0.731·17-s − 0.0106·19-s + (0.0685 − 1.07i)21-s + 1.16·23-s + (0.720 − 0.693i)25-s − 0.903i·27-s − 0.224·29-s − 0.829·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.441979147\)
\(L(\frac12)\) \(\approx\) \(2.441979147\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.07 + 0.836i)T \)
7 \( 1 + (-2.64 - 0.168i)T \)
good3 \( 1 + 1.86iT - 3T^{2} \)
11 \( 1 - 4.26iT - 11T^{2} \)
13 \( 1 + 5.57T + 13T^{2} \)
17 \( 1 - 3.01T + 17T^{2} \)
19 \( 1 + 0.0464T + 19T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 + 1.21T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 + 9.87iT - 37T^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 6.78iT - 47T^{2} \)
53 \( 1 - 4.99iT - 53T^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 + 11.5iT - 61T^{2} \)
67 \( 1 + 6.65T + 67T^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 - 9.67T + 73T^{2} \)
79 \( 1 - 5.90iT - 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 1.51iT - 89T^{2} \)
97 \( 1 - 3.01T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.029847967030714085882250583356, −7.68811756178712041723508059869, −7.49058001633562563419709048966, −6.77782626749810983340817022770, −5.59474888391910917095268967508, −5.08530193910475744269426990451, −4.22275220495026650124993307978, −2.39004671215699742980954028140, −2.03229972512815551719003948065, −0.973034836185128733369634769760, 1.24999406483753572009034742459, 2.57820601247999947467342012575, 3.37791976508443878804273964503, 4.54204275632558080181503448487, 5.19099876416527089374808223650, 5.73486715571273809489469122788, 6.90745671600647214593269986821, 7.64732814928561665955765322779, 8.629243140282795536332125135345, 9.313117651442164895726896965303

Graph of the $Z$-function along the critical line